Sometimes, when Google has one of those seasonal logos I click on it - to see what it’s about. Just now I noticed that the Google logo has a red planet with some little aliens representing the first “O” in Google. Out of curiosity I clicked on it, the image links to Google Mars, which is like Google Maps. Unfortunately it can’t be used with Google Earth client.
The blurb claims these are some of the most detailed maps of Mars ever made. I don’t know if the maps cover the whole planet, as admittedly my Martian geography is pretty poor.
Just for the fun of it I tried moon.google.com and it resolves. Go take a look and make sure to zoom in
The rest of the planets don’t resolve. You can aslo get to Google Mars via mars.google.com.
I seem to get a fair number of hits querying the basic formulas for velocity, so I’ve decided to post them.
Let [tex]\cal{O}[/tex] be an object moving along a path. The position of the object [tex]\cal{O}[/tex] is a function of the time [tex]t[/tex]. We denote the position vector [tex]\bold{x}(t)[/tex].
The time derivative of the position vector [tex]\bold{x}(t)[/tex] is the velocity vector [tex]\bold{v}(t)[/tex] of the object [tex]\cal{O}[/tex].
[tex]\bold{v}(t) \equiv \frac{d\bold{x}(t)}{dt} = \bold{\dot{x}}(t)[/tex].
The time derivative of the velocity vector [tex]\bold{v}(t)[/tex] is called the acceleration vector [tex]\bold{a}(t)[/tex] of the object [tex]\cal{O}[/tex] .
[tex]\bold{a}(t) \equiv \frac{d\bold{v}(t)}{dt} = \frac{d^2\bold{x}(t)}{dt^2} = \bold{\ddot{x}}(t) [/tex].
Integrate this last equation with respect to [tex]t[/tex],
[tex]\int \bold{\ddot{x}}(t) \ dt=\int \bold{a}(t) \ dt \implies \bold{\dot{x}}(t) =\bold{a}( t)t + \bold{C}[/tex],
where [tex]\bold{C}[/tex] is a constant vector to be determined by the initial conditions. Now, [tex]\bold{\dot{x}}(t) =\bold{v}(t)[/tex] so that
[tex]\bold{v}(t) =\bold{a}(t) t + \bold{C}[/tex]
We assume that at time [tex]t =0[/tex] the object [tex]\cal{O}[/tex] is moving with initial velocity [tex]\bold{v}(0) =\bold{v_0}[/tex]
[tex] \bold{v}(0) = \bold{v_0} =\bold{a}(0) \ 0 + C \implies C = \bold{v_0}[/tex]
and substitution gives the familiar formula for the velocity of an object
[tex]\bold{v}(t) = \bold{v_0} + \bold{a}(t) t[/tex].
We can integrate the equation
[tex]\bold{\dot{x}}(t) =\bold{v_0} + \bold{a}(t) t[/tex] with respect to [tex]t[/tex] a second time
[tex]\int \bold{\dot{x}}(t) \ dt = \int \bold{v_0} + \bold{a}(t) t \ dt[/tex] to give
[tex] \bold{x}(t) = \bold{v_0} t+ \frac{1}{2}\bold{a}(t) t^2 + \bold{K} [/tex], where [tex]\bold{K}[/tex] is a constant vector to be determined from the initial conditions.
We assume that at [tex]t =0[/tex] the object [tex]\cal{O}[/tex] has position vector [tex]\bold{x}(0)=\bold{x_0}[/tex]
[tex] \bold{x}(0) = \bold{x_0} = \bold{v_0} t+ \frac{1}{2}\bold{a}(0) 0^2 + \bold{K} \implies \bold{K}=\bold{x_0}[/tex]
and we have the familiar formula for the displacement (position) of the object at time [tex]t[/tex]
[tex] \bold{x}(t) = \bold{x_0} + \bold{v_0} t+ \frac{1}{2}\bold{a}(t) t^2 [/tex].
In a first treatment we often simplify things a bit. Firstly, to simplify notation we don’t write in the explicit time dependence for the position [tex]\bold{x}(t)[/tex] , so that it becomes [tex]\bold{x}[/tex] and similarly for [tex]\bold{v}[/tex] and [tex]\bold{a}[/tex]. Secondly we assume that the object starts from the origin rather than from some arbitrary position, thus we take [tex]\bold{x_0}=0[/tex], (which is really just a change of co-ordinates). This gives the familiar form of the equations of motion from high school text books:
[tex] \bold{x} = \bold{v_0} t+ \frac{1}{2}\bold{a} t^2 [/tex]
[tex]\bold{v} = \bold{v_0} + \bold{a} t[/tex].
These equations are commonly written as
[tex] s = u t+ \frac{1}{2} a t^2 [/tex]
[tex] v = u+ a t[/tex],
where we have used the alternate notation [tex]s \equiv x[/tex] for the position and [tex]u \equiv v_0[/tex] for the intial velocity. Note that we drop the bold face symbols [tex]\bold{v} \rightarrow v[/tex] which are used to denote vectors, since in a first treatment we consider only the one dimensional or scalar case.
Squaring the last equation gives
[tex]v^2 = (u+ a t)^2 = u^2 + 2 u a t + a^2t^2 = u^2 + 2a (ut+\frac{1}{2}at^2) [/tex]
[tex]\phantom{v^2}= u^2 + 2as[/tex]
another commonly used equation in high school physics texts.
The average velocity over the time interval [tex] (t_1,t_2) \equiv (t_i,t_f)[/tex] is given by
[tex]v_{av} = \frac{s_2 -s_1}{t_2-t_1} = \frac{s_f-s_i}{t_f-t_i} = \frac{\Delta(s)}{\Delta(t)}[/tex]
where, [tex] s_1 \equiv s_i[/tex] is the intial displacement and [tex]s_2 \equiv s_f[/tex] is the final displacement. Finally, in words: it is the change in position [tex]\Delta(s)[/tex] divided by the change in time [tex]\Delta(t)[/tex].